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In this paper we define associated spherical partner, radial function and radial spherical function for space curve which doesn't lie on the sphere in Euclidean 3-space. Using these notions we discuss some special curves and their associated spherical partners. Especially we give a new necessary and sufficient condition for the rectifying curves in Euclidean 3-space.

A G-map f : X → Y is called isovariant if it preserves isotropy groups (stabilizers), that is, G x = G f ( x ) for all x ∈ X . For a given compact group G we consider the category I s o v - M of metrizable G-spaces and isovariant maps. In a natural way the notion of an isovariant absolute (neighborhood) extensor ( I s o v - A ( N ) E ) can be introduced; this was done in [1] . Using this notion we define the concept of an isovariant fibrant space and study its properties. In particular, we show that every compact metric G-space admits an isovariant fibrant extension.

Recall that a space X is selectively pseudocompact if for every sequence { U n : n ∈ N } of non-empty open subsets of X one can choose a point x n ∈ U n for all n ∈ N such that the resulting sequence { x n : n ∈ N } has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by Garcia-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, Garcia-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

The free topological vector space V ( X ) over a Tychonoff space X is a pair consisting of a topological vector space V ( X ) and a continuous mapping i = i X : X → V ( X ) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f ‾ : V ( X ) → E with f = f ‾ ∘ i . In this paper, the k-property and countable tightness of free topological vector space over some generalized metric spaces are studied. We mainly discuss the characterizations of a space X such that V ( X ) or the fourth level of V ( X ) is a k-space or is of countable tightness, respectively.

For a Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free proper G-spaces which have a paracompact orbit space. Our approach is based on Milnor's construction E G . We show that E G and the universal free G-spaces in question are G-AE's. Besides, we show how to extend these results to the case of arbitrary metrizable, locally compact, almost connected group actions.

Recently, Zhao and Ho introduced and studied SI-continuous spaces, which can be seen as topological counterparts of continuous posets. The main purpose of this paper is to investigate the relationships between SI-continuous spaces and continuous posets. We prove that a C-space is an SI-continuous space if and only if it is a continuous poset under the specialization order. Furthermore, we introduce the notion of strong SI-continuous spaces and construct an adjunction between the category of domains and the category of strong SI-continuous spaces.

We work in set theory without the Axiom of Choice ( AC ) and establish the following results: 1. “Products of ultracompact spaces are ultracompact” + “there exists a free ultrafilter on ω” ( UF ( ω ) ) is equivalent to AC in ZFA ; 2. “Products of ultracompact spaces are ultracompact” does not imply AC in ZF ; 3. AC is equivalent to each of “products of ultracompact spaces are countably compact”, “products of ultracompact spaces are sequentially limit point compact”, “products of ℵ 0 -bounded spaces are countably compact”, “products of ℵ 0 -bounded spaces are sequentially limit point compact”, and “products of ℵ 0 -bounded spaces are ℵ 0 -bounded” in ZFA ; 4. UF ( ω ) is equivalent to each of “every ultracompact space is sequentially limit point compact” and “every ultracompact T 2 space is sequentially limit point compact”; 5. The Axiom of Countable Choice ( AC ω ) is equivalent to “every ℵ 0 -bounded space is countably compact”; 6. AC ω + UF ( ω ) is equivalent to each of “every ultracompact space is countably compact”, “products of countably many ultracompact spaces are countably compact”, “products of countably many ultracompact spaces are sequentially limit point compact”, and “products of countably many ℵ 0 -bounded spaces are ℵ 0 -bounded”; 7. Each of “a T 3 space is ultracompact, if and only if, it is ℵ 0 -bounded” and “a Tychonoff space is ultracompact, if and only if, it is ℵ 0 -bounded” is equivalent to “every filter on ω can be extended to an ultrafilter on ω” ( BPI ( ω ) ), and thus each of the above statements is equivalent to the statement “the Stone space βω of all ultrafilters on ω is compact”. Thus we provide the exact characterizations of the above topological results, the former by Vaughan in “Countably compact and sequentially compact spaces”, and the latter by Bernstein in “A new kind of compactness for topological spaces”, as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF ); 8. AC ω implies Ginsburg's theorem in “Some results on the countable compactness and pseudocompactness of hyperspaces”, namely “a T 2 space X is ultracompact, if and only if, the hyperspace C L ( X ) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact”, which in conjunction with UF ( ω ) implies the weak choice principle A C fin ω ( AC restricted to countable families of non-empty finite sets).

In 1987, Dress and Scharlau studied the well-known concept of gated sets in metric spaces in order to investigate projections in Tits buildings. In this article, we introduce analogously the concept gated sets in a T 0 -quasi-metric space that we call in-gated set. It turns out that in a T 0 -quasi-metric space, there is a dual concept which we call outgated set. We use these concepts to extend some classical results on gluing a family of hyperconvex metric spaces along a set such that the resulting space preserves hyperconvexity from a metric point of view to quasi-metric settings.

The class of weakly linearly Lindelof spaces was introduced and studied by Juhasz, Tkachuk and Wilson in [7] . Recall that a space X is weakly linearly Lindelof if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U . In this paper, we show that: (1) If X is a weakly linearly Lindelof space and U is an open cover of X, then for the open cover { St 2 ( x , U ) : x ∈ X } of X, there exists a countable subset A ⊂ X such that St 2 ( A , U ) ‾ = X ; (2) Every weakly linearly Lindelof normal metaLindelof space is weakly Lindelof; (3) If X is a first countable regular space, then M ( X ) (generated by Moore Machine) is weakly linearly Lindelof if and only if X is weakly linearly Lindelof; (4) Every product of a weakly linearly Lindelof space and a space of countable spread (or a separable space) is weakly linearly Lindelof; (5) If a subspace X ⊂ ω 1 ω is weakly linearly Lindelof, then X is second countable (and hence, metrizable); (6) If X is a weakly linearly Lindelof Baire space with a rank 2-diagonal such that w e ( X ) ≤ ω 1 , then | X | ≤ c ; (7) The space X is cellular-WLL if and only if it is weakly linearly Lindelof.

In this paper, we introduce sheaves on diffeological spaces as sheaves on the site of plots. We also define quasi-sheaves by presheaves that respect the limit over covering generating families. It is shown that every sheaf on diffeological spaces is a quasi-sheaf. Moreover, every sheaf on a diffeological space gives a sheaf on the D-topological structure. Finally, we characterize a class of sheaves, so-called comma sheaves, and prove that it is equivalent to the comma category of diffeological spaces.

Bal and Kocinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and for every sequence ( A n : n ∈ ω ) of maximal pairwise disjoint open families in X, there exists a sequence ( A n : n ∈ ω ) such that A n ∈ A n for every n ∈ ω and St ( ⋃ n ∈ ω A n , U ) = X . In this paper, we show that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelof nor selectively star-ccc, which gives a positive answer to a question of Bal and Kocinac [2] . Under 2 ℵ 0 = 2 ℵ 1 , we even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelof nor selectively star-ccc. Finally, we prove that every open F σ -subset of a selectively star-ccc space is selectively star-ccc. Some new questions are also posed.

In general topology, filters have proved a popular tool for the construction of completions. In pointfree topology, downsets are frequently employed for the same purpose. Indeed, any downset frame can be viewed as the topology of the space of filters on that frame. In this paper we consider such ideas in the setting of partial frames; these are meet-semilattices where, in contrast with frames, not all subsets need have joins; a so-called selection function specifies those subsets that must have joins. We have made use of the lattice of certain kinds of downsets in several situations; to understand the spatial aspect of these downsets, we are led naturally to consider the notion of partial spaces as a generalization of topological spaces. We establish an adjunction between partial frames and partial spaces which restricts to an equivalence between spatial partial frames and sober partial spaces, and then obtain the desired result that the appropriate collection of downsets is isomorphic to the collection of opens of an appropriate filter space. Both the filter space and the downset partial frame construction are functorial; and the two functors are linked by two natural isomorphisms that involve the open and spectrum functors for partial spaces and partial frames. Next we consider questions of spatiality for the congruence frame of a partial frame. Quotients for frames can be provided using nuclei or congruences but in the setting of partial frames nuclei do not suffice, whereas congruences do. Whether the congruence lattice of a partial frame is itself a frame or not depends on the axioms assumed for the selection function under consideration: if at least all finite subsets are selected, the congruence lattice is indeed a frame. For such, we characterize the spatiality of the frame of all congruences on a partial frame in terms of its quotients.

In this paper we prove a metrization theorem on rectangular b-metric spaces. Then we get a sufficient and necessary condition for a rectangular b-metric space to be metrizable.

A space X for which X ω is Lindelof is called powerfully Lindelof. Extending this term, I call a collection of topological spaces powerfully Lindelof if the product of each of its countable subcollections is Lindelof. In 1971 E. Michael showed (CH) that there is no single such class; here I explore what can be said of them. In doing so, I will focus upon some of the techniques used to prove countable products Lindelof.

We prove the following consistency result for cardinal sequences of length ω 3 : if GCH holds and λ ≥ ω 2 is a regular cardinal, then in some cardinal-preserving generic extension 2 ω = λ and for every ordinal η ω 3 and every sequence f = 〈 κ α : α η 〉 of infinite cardinals with κ α ≤ λ for α η and κ α = ω if cf ( α ) = ω 2 , we have that f is the cardinal sequence of some LCS space. Also, we prove that for every specific uncountable cardinal λ it is relatively consistent with ZFC that for every α , β ω 3 with cf ( α ) ω 2 there is an LCS space Z such that CS ( Z ) = 〈 ω 〉 α ⌢ 〈 λ 〉 β .

A space X is said to be Menger if for any sequence { U n : n ∈ ω } of open covers of X, there exist finite V n ⊂ U n ( n ∈ ω ) such that ⋃ { V n : n ∈ ω } is a cover of X. For a zero-dimensional space X, let C p ( X , 2 ) be the space of all { 0 , 1 } -valued continuous functions with the topology of pointwise convergence. Bernal-Santos and Tamariz-Mascarua [4] gave several results when C p ( X , 2 ) is Menger. In this paper, we give some improvements of them.

In this note the ( J ⁎ )-property, being a generalization of the (*)-property, (λ*)-property and ( d ⁎ ) -property, is defined. There is proved that if J is an ideal of subsets of the real line containing all singletons and a topology τ has the ( J ⁎ )-property then the family of Darboux functions continuous with respect to the topology τ in the domain is strongly porous set in the space of all functions having τ-Darboux property with the supremum metric.

Star selection principles first introduced and studied by Kocinac et al. in [10] and [2] are investigated. In particular we consider the relationship between the star-Menger, strongly star-Menger as well as the relationship between star covering properties of the Niemytzki plane with the dominating number and the bounding number. Furthermore, we give a characterization of the strongly star-Menger property in terms of games on a strongly star-Lindelof space which consist of the disjoint union of a closed discrete set with a σ-compact subspace. A consistent example of a metacompact star-Menger not strongly star-Menger space is given. In addition, we show some results related to weaker properties than paracompactness and we study particular examples. We pose several problems about these properties.

O , Ω and Γ denotes the family of all open, open ω- and open γ-covers of a topological space X, respectively. O s h , Ω s h and Γ s h denotes the corresponding families of shrinkable covers. Let Ψ be one of the symbols O , Ω or Γ. We introduce a property ( Ψ 0 ) of a set of real functions on X. Ψ 0 ( F ) is the set of all subsets of a family of real functions F possessing the property ( Ψ 0 ) . Now, let Φ be one of symbols Ω or Γ. Then for any couple 〈 Φ , Ψ 〉 different from 〈 Ω , O 〉 , a normal topological space X is an S 1 ( Φ s h , Ψ ) -space, if and only if the set C p ( X ) satisfies S 1 ( Φ 0 , Ψ 0 ) , i.e., if S 1 ( Φ 0 ( C p ( X ) ) , Ψ 0 ( C p ( X ) ) ) holds true. Similarly, X is an S 1 ( Φ , Ψ ) -space if and only if the set of non-negative upper semicontinuous real functions on X satisfies S 1 ( Φ 0 , Ψ 0 ) .

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we consider status under PFA(S) of two important consequences of PFA concerning spaces of countable tightness. We prove that the existence of a Souslin tree does not imply the existence of a compact non-sequential space of countable tightness. We contrast this with M.E. Rudin's result that the existence of a Souslin tree does imply the existence of an S-space (and the later improvement by Dahrough to a compact S-space). On the other hand, PFA(S) implies there is a first-countable perfect pre-image of ω 1 that contains no copies of ω 1 .

In this paper we use some cardinal functions introduced in [3] , [5] and [1] , and the technique of elementary submodels to generalize some cardinal inequalities. Also, we introduce the cardinal functions, denoted by ( γ , θ ) − w L θ and U θ ⁎ , and we prove that: | X | ≤ 2 ( γ , θ ) − w L θ ( X ) χ ( X ) , for any T 1 -space X with U θ ⁎ ( X ) ≤ ω .

We prove that every cellular-Lindelof space is weakly linearly Lindelof and hence monotonically normal cellular-Lindelof spaces must be Lindelof. We also show that there exists a cellular-Lindelof locally Lindelof P-space that is not weakly Lindelof. Under CH, we establish that every first countable normal weakly linearly Lindelof space is weakly Lindelof and has cardinality not exceeding c . Our results solve an open question published by Bella and Spadaro.

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions.

We give a topological approach to the Ulam–Kakutani–Collatz conjecture. In fact we prove that if ( N , τ f ) is not a w– R 0 space is equivalent to saying that the conjecture is true, τ f being the topology on N given by the open sets as those subsets θ of N such that f − 1 ( θ ) ⊂ θ , where f is the Collatz function.

We study the completeness and ultracompleteness numbers of a convergence space. In the case of a completely regular topological space, the completeness number is countable if and only if the space is $\v{C}$ech-complete, and the ultracompleteness number is countable if and only if the space is ultracomplete. We show that the completeness number of a space is equal to the pseudopaving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$. Similarly, the ultracocompleteness number of a space is equal to the paving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$., Comment: 17 pages

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that: (1) A subspace X ⊂ ω 1 ω is dually CCC then e ( X ) ≤ ω and a normal subspace X ⊂ ω 1 ω is DCCC if and only if e ( X ) ≤ ω ; (2) There is a Tychonoff pseudocompact subspace X ⊂ ( ω 1 + 1 ) 2 which is not dually CCC; (3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH; (4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11] ; (5) Every dually separable and monotonically monolithic space is Lindelof, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1] ; (6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2 c . The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “ G δ -diagonal”; (7) A dually separable ω-monolithic Hausdorff space with a G δ -diagonal has cardinality at most c .

Let H be a subgroup of the fundamental group π 1 ( X , x 0 ) . In this paper, first, we give a positive answer to a problem presented by Brodskiy et al. in [8, Problem 4.8] . Second, by extending the concept of strong SLT space to a relative version with respect to H, strong H-SLT space, we investigate the existence of a covering map for strong H-SLT spaces. Moreover, we show that a semicovering map is a covering map in the presence of strong H-SLT property. Among other things, we present conditions under which the whisker topology agrees with the lasso topology on X ˜ H . Also, we study the relationship between open subsets of π 1 w h ( X , x 0 ) and π 1 l ( X , x 0 ) . Finally, we give several examples to justify the definition and study of strong H-SLT spaces when H is a non-trivial subgroup.

We investigate different concepts related to D-spaces. We show that any space with an ω 0 -Noetherian point network is hereditarily thickly covered. To distinguish monotonically monolithic spaces and spaces with predictable network, we show that there exists a monotonically monolithic space which doesn't have a predictable network under the assumption 2 ω 0 2 ω 1 . This provides a consistent negative answer to one of Guo and Junnila's questions in [16] . Some sufficient conditions for a strongly monotonically monolithic space to have a point-countable base are also given.

We apply the decomposition theory of a manifold, elaborated by Bing and developed by Daverman, to study a decomposition space of a foliated manifold. We consider a minimal homeomorphism preserving foliation on a compact manifold. As a consequence of our main result we obtain a generalization of the result of Kolyada, Snoha and Trofimchuk concerning a minimal skew-product homeomorphism of the 2-dimensional torus.

We introduce the notions of metric finite expansive homeomorphism, orbit finite expansive homeomorphism, and topologically finite expansive homeomorphism on compact metric spaces, topological spaces, and uniform spaces, respectively. These notions coincide on compact metric spaces, but they are not the same on general metric spaces. We give some examples to show that the metric finite expansivity, orbit finite expansivity, and topological finite expansivity are weaker than the metric n-expansivity, orbit expansivity and topological expansivity, respectively. We state suitable conditions to imply that the metric finite expansivity is equal to the metric expansivity. We show that any regular recurrent point of a metric finite expansive homeomorphism f : X → X is a periodic point and if X is an uncountable compact metric space, then Ω ( f ) is an infinite set. It is known that if there is an orbit expansive homeomorphism on X, then X is a T 1 -space, in spite of it, we give an orbit finite expansive homeomorphism f : X → X such that X is not a T 1 -space. Then we show that if f : X → X is an orbit finite expansive homeomorphism with the orbit finite expansive covering { U i } i = 1 n , then X − U i is an infinite set for all 1 ≤ i ≤ n . Finally we show that if f : X → X is a homeomorphism on a compact Hausdorff space X, then f has a weak finite generator if and only if f is a topologically finite expansive homeomorphism.

For a given space X let L ( X ) be the family of all compact subsets of X. A space X is dominated by the space M if X has an M-ordered compact cover, which means that there exists a family F = { F K : K ∈ L ( M ) } ⊂ L ( X ) such that ⋃ F = X and K ⊂ L implies that F K ⊂ F L , whenever K , L ∈ L ( M ) . A space X is strongly dominated by a space M if it has an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . A space X is called ω-hyperbounded if the closure of any σ-compact subspace of X is compact. We prove that if a space X is (strongly) dominated by an ω-hyperbounded space, then it is (ω-hyperbounded) ω-bounded. We prove that, for a given cardinal κ, if a space X is (strongly) dominated by a κ-hemicompact space, then it is (κ-hemicompact) κ-compact. We analyze relation between domination and hyperspaces of nonempty compact subsets.

This manuscript extends both Cantor intersection theorem and Cantor–Kuratowski intersection theorem from the setting of metric spaces to the setting of admissible spaces. Completeness is revisited and the notion of uniformly locally compact admissible space is introduced. Compact, complete, and totally bounded admissible spaces are related.

Given a Lindelof space X and a regular uncountable cardinal κ, we establish that κ is a caliber of C p ( X ) if and only if the diagonal of the space X is κ-small. We show that the Lindelof property of X cannot be omitted in this result because there exists a pseudocompact space X with a G δ -diagonal such that ω 1 is not a caliber of C p ( X ) . We also construct an example of a pseudocompact space X such that ω 1 is a caliber of C p ( X ) but not a caliber of C p ( β X ) = C p ( υ X ) ; this solves two published open questions.

We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data., Changes from last version: removed odd parity assumption in result on classes in higher-dimensional spaces of link maps; other minor revisions. 23 pages. Accepted for publication in Topology Appl

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n-2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability., Replaces an earlier version called "The tangent bundle of planar polygon spaces." Adds a good result about parallelizability

Let I = [ 0 , 1 ] with b B 1 the set of Baire-1 self-maps of I. There exists S a residual subset of b B 1 such that for any f ∈ S , the following hold: 1. The set of the ω-limit points Λ ( f ) = ∪ x ∈ I ω ( x , f ) is a nowhere dense and perfect subset of [ 0 , 1 ] with Hausdorff dimension zero. 2. The collection of the ω-limit sets Ω ( f ) = { ω ( x , f ) : x ∈ I } generated by f is closed in the Hausdorff metric space. 3. If x is a point at which f is continuous, then ( x , f ) is a point at which the map ω : I × b B 1 → K given by ( x , f ) → ω ( x , f ) is continuous. 4. The n-fold iterate f n is an element of b B 1 for all natural numbers n.

By F n ( X ) , n ≥ 1 , we denote the n-th symmetric product of a metric space ( X , d ) as the space of the nonempty finite subsets of X with at most n elements endowed with the Hausdorff metric d H . By Iso ( X ) we denote the group of all isometries from X onto itself with the topology of pointwise convergence. In this paper, we show that, under the certain hypothesis, Iso ( F n ( X ) ) is topologically isomorphic to the semidirect product group Iso ( F n ( X ) , F 1 ( X ) ) ⋊ Iso ( X ) . We apply those results to l p q , ( p , q ) ∈ [ 1 , ∞ ] × N ≥ 2 ⁎ , as particular spaces and prove the following statements: (1) If p ∈ { 1 , ∞ } , then Iso ( F 2 ( l p 2 ) ) is topologically isomorphic to Z 2 × Iso ( l p 2 ) . (2) If 3 ≤ q ∞ , then Iso ( F 2 ( l ∞ q ) ) is topologically isomorphic to ∏ i = 1 q − 1 ( Z 2 ) i ⋊ Iso ( l ∞ q ) . (3) In other cases except ( n , p , q ) ∈ N ≥ 2 × { 1 , ∞ } × { ∞ } , the canonical homomorphism χ n : Iso ( l p q ) → Iso ( F n ( l p q ) ) is a topological isomorphism.

Let Y be an ordered topological vector space with a positive interior point e. Motivated by the Minkowski functional p e , we introduce the function q e on the interior int Y Y + of the positive cone of Y. The function q e plays a role for evaluating a ‘reverse gauge’ of the Minkowski functional p e on the restricted space int Y Y + . We also apply the function q e for giving complete proof of the recent results [7] by Jin, Xie and Yue on monotonically countably paracompact spaces. New characterizations by vector-valued continuous maps are also given for stratifiable spaces and monotone cb-spaces, the latter provides a positive answer to a question posed in [12] by the author.

Let X be a topological space, ( Y , d ) be a metric space, Q ( X , Y ) be the space of quasicontinuous functions from X to Y and τ U C be the topology of uniform convergence on compacta. We study first countability, metrizability and complete metrizability of ( Q ( X , Y ) , τ U C ) . We will apply our results to characterize sequentially compact subsets of ( Q ( X , Y ) , τ U C ) .

A space Z is weakly pseudocompact if Z is G δ -dense in at least one of its compactifications. In 1996 F.W. Eckertson [3] proposed the following problem: Find examples of Baire non Lindelof spaces which are not weakly pseudocompact. Eckertson proposed a list of natural candidates. In this article we show that part of this list produces examples of this type by providing examples of product spaces which are Baire non-Lindelof and not weakly pseudocompact.

A space X is said to be quasi-Rothberger if for each closed set F ⊂ X and each sequence { U n : n ∈ N } of covers of F by sets open in X, there is a U n ∈ U n for each n ∈ N such that F ⊂ ⋃ n ∈ N U n ‾ . In this article, we give necessary and sufficient conditions of the quasi-Rothberger property of linearly ordered spaces.

We prove that a Hausdorff metaLindelof weakly star countable space is feebly Lindelof and a Hausdorff metacompact weakly star finite space is almost compact which partially answers a question of Alas and Wilson (2017) [2, Question 3.14] . We also obtain a normal example of a weakly star countable space which is neither almost star countable nor star Lindelof without any set-theoretic assumptions, which answers a question implicitly asked by Song (2015) [13, Remark 2.8] and a question asked by Alas, Junqueira and Wilson (2011) [3, Question 4] . Under MA+¬CH, there even exists a normal weakly star countable Moore space which is not almost star countable. An example of a Tychonoff star compact and weakly star finite space which is not star countable is also given. Finally, we prove that every weakly star countable Hausdorff space with a rank 4-diagonal has cardinality at most 2 ω .

A variation of the proximal infinite game and a class of spaces more general than the proximal spaces are introduced. If the first player has a winning strategy in this variation then the space is pseudonormal. If the second player does not have a winning strategy then the space is an Arhangel'skii α 2 space. This new version of the proximal game is used to show that a Σ-product of ω -bounded topological manifolds is pseudonormal.

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product X J is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT ( fin ) are deduced. It is shown that, in every model of ZF + ¬ CUT ( fin ) , for an uncountable set J, the space N J can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

In this paper, after discussing the one point compactification of the Khalimsky line (resp. the Khalimsky plane), denoted by ( Z ⁎ , κ ⁎ ) (resp. ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) ), we study various properties of these compactifications associated with the semi- T 1 2 axiom, a non-Alexandroff structure, a non-cut-point space and so forth. We also investigate dense subsets and nowhere dense subsets of ( Z ⁎ , κ ⁎ ) and ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) . Finally, motivated by a particular point topology and an excluded point topology, we develop two kinds of new topologies as quotient topological spaces of ( Z ⁎ , κ ⁎ ) called an excluded two points topology and a cofinite particular point topology.

A space X is selectively absolutely star-Lindelof [1] , [3] if for each open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X, there are finite sets F n ⊆ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . In this paper, we continue to investigate topological properties of selectively absolute star-Lindelof spaces, and show the following statements: (1) There exists a Tychonoff selectively a-star-Lindelof, pseudocompact space X having a regular closed G δ subset which is not star-Lindelof (hence not selectively a-star-Lindelof); (2) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal selectively a-star-Lindelof space X having a regular closed G δ subset which is not star-Lindelof (hence not selectively a-star-Lindelof); (3) An open F σ -subset of a selectively a-star-Lindelof space is selectively a-star-Lindelof; (4) For any cardinal κ, there exists a Tychonoff selectively a-star-Lindelof, pseudocompact space X such that e ( X ) ≥ κ .

A space X is called a k R -space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we discuss the k R -property of products of sequential fans and free Abelian topological groups by applying the κ -fan introduced by Banakh. In particular, we prove the following two results: (1) The space S ω 1 × S ω 1 is not a k R -space. (2) The space S ω × S ω 1 is a k R -space if and only if S ω × S ω 1 is a k -space if and only if b > ω 1 . These results generalize some well-known results on sequential fans. Furthermore, we generalize some results of Yamada on the free Abelian topological groups by applying the above results. Finally, we pose some open questions about the k R -spaces.

Inverse limit spaces of compacta with upper semi-continuous compact set-valued functions are studied via shift maps and their variants. We give a representation of such spaces as the limits of ordinary inverse sequences, which allows us to prove some known results and their extensions in a unified scheme. Next we present a scheme to construct shift dynamics on the inverse limit space with various dynamical features. In particular we construct an inverse sequence over [ 0 , 1 ] with a single upper semi-continuous function f as its bonding function such that (i) the inverse limit space [ 0 , 1 ] f is homeomorphic to the Cantor set and (ii) the shift map σ f : [ 0 , 1 ] f → [ 0 , 1 ] f is topologically conjugate to a minimal subshift of a Bernoulli full shift. Also we study local/global connectivity of the inverse limit space over a compactum with a single upper semi-continuous bonding function in terms of homotopy/(co)homology groups, again via shift maps and their variants.

A Lagrangian star is a system of three Lagrangian submanifolds of the symplectic space intersecting at a common point. In this work we classify transversal Lagrangian stars in the symplectic space in the analytic category under the action of symplectomorphisms by using the method of algebraic restrictions. We present a list of all transversal Lagrangian star.

The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G × G into G is sequentially continuous. In this paper, we mainly consider the properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological groups. It is showed that (1) every snf -countable semitopological group G with the sequentially continuous multiplication is sof -countable; (2) if G is a sequential quasitopological group with the sequentially continuous multiplication, then G contains a closed copy of S ω if and only if it contains a closed copy of S 2 , which give a partial answer to a problem posed by R.-X. Shen; (3) let G be a quasitopological group with the sequentially continuous multiplication, then the following are equivalent: (i) G is a sequential α 4 -space; (ii) G is Frechet; (iii) G is strongly Frechet; (4) (MA+¬CH) there exists a non-metrizable, separable, normal and Moore quasitopological group; (5) some examples are constructed to show that metrizability, first-countability and second-countability are not three-space properties in the class of quasitopological groups.